Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2023 Summer Zone 3

(a) Translate triangle $A$ by $\begin{pmatrix} -6 \\ 4 \end{pmatrix}$. Label the image $C$. [2]
(b) Describe fully the \textit{single} transformation that maps triangle $A$ onto triangle $B$.
.....................................................................................................................................
..................................................................................................................................... [3]
(c) Reflect triangle $A$ in the line $x = -2$. Label the image $D$. [2]
(d) Enlarge triangle $A$ by scale factor $-1$, centre $(0, 0)$. Label the image $E$. [2]

A triathlon race consists of three parts:
• a 1500 m swim
• a 40 km bike ride
• a 10 km run.

(a) John swims the 1500 m in 25 minutes.
Find his average speed, in km/h, for this swim.
......................................... km/h [2]

(b) John completes the 40 km bike ride at an average speed of 32 km/h.
Find the time, in minutes, for John to complete this bike ride.
......................................... min [2]

(c) John completes the whole race at an average speed of 20.6 km/h.
Find the average speed, in km/h, for John to complete his 10 km run.
......................................... km/h [3]

The table shows the marks of 12 students in a French examination and a Spanish examination.

\[\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \text{French mark (x)} & 17 & 23 & 28 & 32 & 37 & 42 & 57 & 61 & 77 & 82 & 94 & 96 \\ \hline \text{Spanish mark (y)} & 26 & 22 & 33 & 46 & 41 & 53 & 62 & 67 & 66 & 75 & 83 & 95 \\ \hline \end{array}\]

(a) Find the \textbf{median} Spanish mark.
....................................................... [1]

(b) Find the mean French mark.
....................................................... [1]

(c) Find the equation of the regression line for \( y \) in terms of \( x \).
\( y = \) ....................................................... [2]

(d) Use your equation to estimate the Spanish mark when
(i) the French mark is 50
....................................................... [1]
(ii) the French mark is 6.
....................................................... [1]

(e) Which French mark, 50 or 6, is likely to give the most reliable Spanish mark?
Give a reason for your answer.
.................... because ...............................................................................................
....................................................... [1]

(a) $x$ is divided in the ratio 3 : 5. The larger share is $42$.
Find the value of $x$.

$x =$ ...................................................

(b) (i) Increase 124 by 16%.

................................................... [2]

(ii) The price of a coat is reduced by $\frac{2}{9}$ in a sale.
The new price of the coat is $73.50$.
Find the original price of the coat.

$ ................................................... [2]

(c) Xiong invests $2000 in Bank $A$ which pays simple interest at a rate of 3% each year.
Find the total amount of interest Xiong receives at the end of 5 years.

$ ...................................................

(d) Wendi invests $400 in Bank $B$ which pays compound interest at a rate of 1.6% each year.
Find the total amount of interest Wendi receives at the end of 3 years.

$...................................................

(e) Pedro invests $1000 in Bank $C$ for 18 years.
Pedro also invests $1000 in Bank $D$ for 18 years.
Bank $C$ pays simple interest at a rate of $x\%$ each year.
Bank $D$ pays compound interest at a rate of 0.7% each year.
At the end of 18 years Pedro has exactly the same amount of money in Bank $C$ and Bank $D$.
(i) Show that $1 + \frac{18x}{100} = \left(1 + \frac{0.7x}{100}\right)^{18}$ .

[2]
(ii) Given that $5 < x < 7$, use a graphical method to find $x$.



$x =$ ................................................... [2]

In the diagram, $AB$ is parallel to $ED$.
$ACD$ and $BCE$ are straight lines.
$AB = 50\, \text{cm}$, $BC = 55\, \text{cm}$ and $AC = 64\, \text{cm}$.

(a) Show that angle $ACB = 49.0^\circ$ correct to one decimal place.
(b) Use the sine rule to calculate angle $CAB$.
Angle $CAB = \text{................................................}$ [3]
(c) Calculate the area of triangle $ABC$.
........................................... $\text{cm}^2$ [2]
(d) $AC = \frac{2}{3}AD$
Calculate the area of triangle $CDE$.
......................................... $\text{cm}^2$ [2]

(a) Solve.
$$7x - 5 = 3x + 13$$
$$x = \text{...........................................} \; [2]$$

(b) Solve.
$$4(2x - 3) = 3(1 - 2x)$$
$$x = \text{...........................................} \; [3]$$

(c) Solve.
$$\frac{3x+2}{8} = \frac{2}{3x+2}$$
$$x = \text{................. or \hspace{2mm} x = .................} \; [3]$$

(d) Solve.
$$1 - 2x^2 = 5x - 1$$
Give your answer correct to two decimal places.
$$x = \text{................. or \hspace{2mm} x = .................} \; [3]$$

(e) $$\log x = 1 + 4\log y$$
Find $x$ in terms of $y$.
$$x = \text{...........................................} \; [3]$$

(f) There are 12 balls in a bag, $n$ of them are blue.
A ball is taken from the bag at random and replaced.
The probability that the ball is blue is $p$.

6 more blue balls are added to the bag.
A ball is taken from the bag at random.
The probability that this ball is blue is $2p$.

Find the value of $p$.
$$p = \text{...........................................} \; [4]$$

f(x) = |2 \cos(x - 45)^\circ| - 1\text{ for values of } x \text{ between } -90 \text{ and } 90.

(a) On the diagram, sketch the graph of \( y = f(x) \). [3]
(b) Write down the \( x \)-coordinates of the points where the curve meets the \( x \)-axis.
\( x = \text{.............. or } x = \text{..............} \) [2]
(c) Write down the coordinates of the local maximum point.
\((\text{..............}, \text{..............})\) [1]
(d) The line \( y = 0.005x \) intersects the curve \( y = |2 \cos(x - 45)^\circ| - 1 \) three times.
(i) Find the \( x \)-coordinates of the points of intersection.
\( x = \text{.............. or } x = \text{.............. or } x = \text{..............} \) [3]
(ii) Solve the inequality.
\(|2 \cos(x - 45)^\circ| - 1 > 0.005x\)
\(\text{..............................................................}\)[2]

(a) Solve the simultaneous equations.

$$5x - 4y = 13$$
$$3x + 2y = -1$$

You must show all your working.

x = ........................................................
y = ........................................................ [3]

(b) $$f(x) = 3x + 1$$ \(\ \) $$g(x) = \frac{1}{2x-3}$$ , $$x \neq 1.5$$

(i) Find \( f(-2) \).

.............................................................. [1]

(ii) Find \( f(f(x)) \), giving your answer in its simplest form.

.............................................................. [2]

(iii) Solve \( g(f(x)) = \frac{1}{5} \).

x = ........................................................ [3]

There are 80 students in a school year, 44 boys and 36 girls. Each student chooses their favourite sport. The number of boys and the number of girls choosing each sport is shown in the table.

[Table_1]

(a) A student is chosen at random from the 80 students.

Find the probability that the student chosen is

(i) a girl whose favourite sport is athletics

...................................................... [1]

(ii) a boy whose favourite sport is not football.

...................................................... [1]

(b) One of the girls is chosen at random.

Find the probability that her favourite sport is hockey.

...................................................... [2]

(c) Three of the boys are chosen at random.

(i) Find the probability that one of the boys chooses athletics, one of them chooses football and the other chooses swimming.

...................................................... [1]

(ii) Calculate the probability that the three boys each have a different favourite sport.

...................................................... [3]

A, B, C \, \text{and} \, D \text{ lie on a circle, centre } O.
DE \text{ is a tangent to the circle at } D.
ACE \text{ is a straight line.}
Find
(a) \text{ angle } AOC
\text{Angle } AOC = \text{........................................}\ [1]
(b) \text{ angle } OAC
\text{Angle } OAC = \text{..........................................}\ [2]
(c) \text{ angle } ADC
\text{Angle } ADC = \text{..........................................}\ [1]
(d) \text{ angle } CAD.
\text{Angle } CAD = \text{..........................................}\ [3]

The diagram shows a cuboid with base $ABCD$.
$AB = 20 \, \text{cm}$, $BC = 34 \, \text{cm}$ and $CE = 16 \, \text{cm}$.
Water is poured into the cuboid to a height of $8 \, \text{cm}$.
(a) Find the volume of water in the cuboid. $\text{....................................... cm}^3$ [2]

(b) A sphere of radius $4 \, \text{cm}$ is placed so that it rests on the base of the cuboid.
The water level is now $q \, \text{cm}$ above the base of the cuboid.
Find the value of $q$. [4]

(c) The sphere is removed from the cuboid.
15 identical cubes of side $x \, \text{cm}$ are placed so that they rest on the base of the cuboid.
(i) Find the maximum value of $x$.
$x = \text{.......................................}$ [3]
(ii) The water level is now $p \, \text{cm}$ above the base of the cuboid.
Find the maximum value of $p$.
$p = \text{.......................................}$ [3]

The diagram shows two right-angled triangles $ABC$ and $CBD$.
$AB = BC$, $AC = \sqrt{200}\, \text{cm}$ and angle $BDC = 30^\circ$.
Find the perimeter of triangle $ACD$.